Ordinary and generalized Langevin equation

Let us first explain what we mean here for generalized Langevin equation (GLE). For this, we summarize rather well-known concepts of the theory of Brownian motion and thermal fluctuations cast as stochastic processes, generated by linear stochastic equations with additive noise. 

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The present analysis builds directly on three previous analyses of SDEs for constrained systems by Fixman [9], Hinch [10], and Ottinger [11]. Fixman and Hinch both considered an interpretation of the inertialess Langevin equation as a limit of an ordinary differential equation with a finite, continuous random force. Both authors found that, to obtain the correct drift velocity and equilibrium distribution, it was necessary to supplement forces arising from derivatives of C/eff = U — kT n by an additional corrective pseudoforce, but obtained inconsistent results for the form of the required correction force. Ottinger [11] based his analysis on an Ito interpretation of SDEs for both generalized and Cartesian coordinates, and thereby obtained results that... 

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